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Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2015年第1期167卷 102-117页

作者： Rebennack, Steffen Kallrath, Josef Colorado Sch Mines Div Econ & Business Golden CO 80401 USA Univ Florida Dept Astron Gainesville FL 32611 USA BASF SE Sci Comp D-67056 Ludwigshafen Germany

For functions depending on two variables, we automatically construct triangulations subject to the condition that the continuous, piecewise linear approximation, under-, or overestimation, never deviates more than a given -tolerance from the original function over a given domain. This tolerance is ensured by solving subproblems over each triangle to global optimality. The continuous, piecewise linear approximators, under-, and overestimators, involve shift variables at the vertices of the triangles leading to a small number of triangles while still ensuring continuity over the entire domain. For functions depending on more than two variables, we provide appropriate transformations and substitutions, which allow the use of one- or two-dimensional -approximators. We address the problem of error propagation when using these dimensionality reduction routines. We discuss and analyze the trade-off between one-dimensional (1D) and two-dimensional (2D) approaches, and we demonstrate the numerical behavior of our approach on nine bivariate functions for five different -tolerances.

关键词： Global optimization nonlinear programming mixed-integer nonlinear programming Nonconvex optimization Error propagation

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FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

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MATHEMATICAL programming 2008年第2期115卷 273-290页

作者： De Loera, Jesus A. Hemmecke, Raymond Koeppe, Matthias Weismantel, Robert Otto VonGuericke Univ Magdegurg FMA IMO D-39106 Magdeburg Germany Univ Calif Davis Dept Math Davis CA 95616 USA

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.

关键词： mixed-integer nonlinear programming integer programming in fixed dimension computational complexity approximation algorithms FPTAS

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An Outer-Inner Approximation for Separable mixed-integer nonlinear Programs

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INFORMS JOURNAL ON COMPUTING 2014年第1期26卷 31-44页

作者： Hijazi, Hassan Bonami, Pierre Ouorou, Adam Natl ICT Australia Canberra ACT 2601 Australia Univ Marseille Fac Sci Luminy Lab Informat Fondamentale F-13009 Marseille France Orange Labs R&D CORE TPN F-92794 Issy Les Moulineaux 9 France

A common structure in convex mixed-integer nonlinear programs (MINLPs) is separable nonlinear functions. In the presence of such structures, we propose three improvements to the outer approximation algorithms. The first improvement is a simple extended formulation, the second is a refined outer approximation, and the third is a heuristic inner approximation of the feasible region. As a side result, we exhibit a simple example where a classical implementation of the outer approximation would take an exponential number of iterations, whereas it is easily solved with our modifications. These methods have been implemented in the open source solver BONMIN and are available for download from the Computational Infrastructure for Operations Research project website. We test the effectiveness of the approach on three real-world applications and on a larger set of models from an MINLP benchmark library. Finally, we show how the techniques can be extended to perspective formulations of several problems. The proposed tools lead to an important reduction in computing time on most tested instances.

关键词： mixed-integer nonlinear programming outer approximation

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Cutting planes for RLT relaxations of mixed 0-1 polynomial programs

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MATHEMATICAL programming 2015年第2期151卷 639-658页

作者： Fomeni, Franklin Djeumou Kaparis, Konstantinos Letchford, Adam N. Univ Lancaster Sch Management Dept Management Sci Lancaster LA1 4YX England

The reformulation-linearization technique, due to Sherali and Adams, can be used to construct hierarchies of linear programming relaxations of mixed 0-1 polynomial programs. As one moves up the hierarchy, the relaxations grow stronger, but the number of variables increases exponentially. We present a procedure that generates cutting planes at any given level of the hierarchy, by optimally weakening linear inequalities that are valid at any given higher level. Computational experiments, conducted on instances of the quadratic knapsack problem, indicate that the cutting planes can close a significant proportion of the integrality gap.

关键词： Polynomial optimisation Cutting planes mixed-integer nonlinear programming Quadratic knapsack problem

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Voltage and Reactive Power Optimization Using a Simplified Linear Equations at Distribution Networks with DG

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ENERGIES 2020年第13期13卷 3334-3334页

作者： Go, Seok-Il Yun, Sang-Yun Ahn, Seon-Ju Choi, Joon-Ho Chonnam Natl Univ Dept Elect Engn Gwangju 61186 South Korea

In this paper, the VVO (Volt/Var optimization) is proposed using simplified linear equations. For fast computation, the characteristics of voltage control devices in a distribution system are expressed as a simplified linear equation. The voltage control devices are classified according to the characteristics of voltage control and represented as the simplified linear equation. The estimated voltage of distribution networks is represented by the sum of the simplified linear equations for the voltage control devices using the superposition principle. The voltage variation by the reactive power of distributed generations (DGs) can be expressed as the matrix of reactance. The voltage variation of tap changing devices can be linearized into the control area factor. The voltage variation by capacitor banks can also be expressed as the matrix of reactance. The voltage equations expressed as simplified linear equations are formulated by quadratic programming (QP). The variables of voltage control devices are defined, and the objective function is formulated as the QP form. The constraints are set using operating voltage range of distribution networks and the control ranges of each voltage control device. In order to derive the optimal solution, mixed-integer quadratic programming (MIQP), which is a type of mixed-integer nonlinear programming (MINLP), is used. The optimal results and proposed method results are compared by using MATLAB simulation and are confirmed to be close to the optimal solution.

关键词： Volt Var optimization distributed generation OLTC (On Load Tap Changer) mixed-integer nonlinear programming quadratic programming

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Energy-Optimal Multi-Goal Motion Planning for Planar Robot Manipulators

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第1期163卷 80-104页

作者： Bonami, Pierre Olivares, Alberto Staffetti, Ernesto Ctr Natl Rech Sci Paris France Univ Rey Juan Carlos Madrid Spain

In this work, the energy-optimal motion planning problem for planar robot manipulators with two revolute joints is studied, in which the end-effector of the robot manipulator is constrained to pass through a set of waypoints, whose sequence is not predefined. This multi-goal motion planning problem has been solved as a mixed-integer optimal control problem in which, given the dynamic model of the robot manipulator, the initial and final configurations of the robot, and a set of waypoints inside the workspace of the manipulator, one has to find the control inputs, the sequence of waypoints with the corresponding passage times, and the resulting trajectory of the robot that minimizes the energy consumption during the motion. The presence of the waypoint constraints makes this optimal control problem particularly difficult to solve. The mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the waypoints part of the state, then introducing binary variables to enforce the constraint of passing once through each waypoint, and finally applying a fifth-degree Gauss-Lobatto direct collocation method to tackle the dynamic constraints. High-degree interpolation polynomials allow the number of variables of the problem to be reduced for a given numerical precision. The resulting mixed-integer nonlinear programming problem has been solved using a nonlinear programming-based branch-and-bound algorithm specifically tailored to the problem. The results of the numerical experiments have shown the effectiveness of the approach.

关键词： Multi-goal motion planning Robot manipulators mixed-integer optimal control problem mixed-integer nonlinear programming

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A generalized superstructure-based framework for process synthesis

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COMPUTERS & CHEMICAL ENGINEERING 2020年 133卷

作者： Ryu, Joonjae Kong, Lingxun de Lima, Arthur E. Pastore Maravelias, Christos T. Univ Wisconsin Dept Chem & Biol Engn 1415 Engn Dr Madison WI 53706 USA Univ Wisconsin DOE Great Lakes Bioenergy Res Ctr 1552 Univ Ave Madison WI 53726 USA

We propose a general framework for the formulation of superstructure-based optimization models for holistic process synthesis. First, we redefine the fundamental problems of reactor, separation, and heat exchanger network synthesis, by presenting generalized problem statements, to make them amenable to seamless integration with each other. Second, we describe the general forms of models that can be developed to address these generalized problems and identify some key characteristics. Notably, for each system, we identify internal variables used only within the system, cost variables used in the objective function, and, importantly, coupling variables for the coupling between systems. Third, we outline some literature models that can be used to address the generalized problems and present new models to couple the three systems. Finally, we show how the individual components (systems and coupling models) can be integrated to formulate a single simultaneous reactor, separation, and heat exchanger network synthesis. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Global optimization mixed-integer nonlinear programming Reactors Separations Heat integration Process design

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Minotaur: a mixed-integer nonlinear optimization toolkit

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MATHEMATICAL programming COMPUTATION 2021年第2期13卷 301-338页

作者： Mahajan, Ashutosh Leyffer, Sven Linderoth, Jeff Luedtke, James Munson, Todd Indian Inst Technol Ind Engn & Operat Res Mumbai 400076 Maharashtra India Argonne Natl Lab Math & Comp Sci Div Lemont IL 60439 USA Univ Wisconsin Dept Ind & Syst Engn Madison WI 53706 USA

We present a flexible framework for general mixed-integer nonlinear programming (MINLP), called Minotaur, that enables both algorithm exploration and structure exploitation without compromising computational efficiency. This paper documents the concepts and classes in our framework and shows that our implementations of standard MINLP techniques are efficient compared with other state-of-the-art solvers. We then describe structure-exploiting extensions that we implement in our framework and demonstrate their impact on solution times. Without a flexible framework that enables structure exploitation, finding global solutions to difficult nonconvex MINLP problems will remain out of reach for many applications.

关键词： mixed-integer nonlinear programming Global optimization Software

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SOLVING THE nonlinear DISCRETE TRANSPORTATION PROBLEM BY MINLP OPTIMIZATION

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TRANSPORT 2014年第1期29卷 1-11页

作者： Klansek, Uros Univ Maribor Fac Civil Engn Maribor Slovenia

The nonlinear Discrete Transportation Problem (NDTP) belongs to the class of the optimization problems that are generally difficult to solve. The selection of a suitable optimization method by which a specific NDTP can be appropriately solved is frequently a critical issue in obtaining valuable results. The aim of this paper is to present the suitability of five different mixed-integer nonlinear programming (MINLP) methods, specifically for the exact optimum solution of the NDTP. The evaluated MINLP methods include the extended cutting plane method, the branch and reduce method, the augmented penalty/outer-approximation/equality-relaxation method, the branch and cut method, and the simple branch and bound method. The MINLP methods were tested on a set of NDTPs from the literature. The gained solutions were compared and a correlative evaluation of the considered MINLP methods is shown to demonstrate their suitability for solving the NDTPs.

关键词： transportation problems discrete transporting flows nonlinear cost functions optimization methods mixed-integer nonlinear programming

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A Superstructure-Based Lignin Valorization Process Optimization Model for Lignocellulosic Biorefineries through Biological Upgrading

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ACS SUSTAINABLE CHEMISTRY & ENGINEERING 2024年第18期12卷 6913-6926页

作者： Wu, Yajie Restrepo-Florez, Juan Manuel Vasco-Correa, Juliana Penn State Univ Dept Agr & Biol Engn University Pk PA 16802 USA Univ Florida Dept Chem Engn Gainesville FL 32611 USA

This study presents an optimization framework to identify economically viable pathways for lignin valorization in biorefineries that employ biological upgrading. The economic potential for converting lignin from hardwood, softwood, and herbaceous plants into valuable bioproducts is evaluated. The research indicates that the production of 2-pyrone-4,6-dicarboxylic acid (PDC) from hardwood is the most economically promising, with an estimated net present value of $771.41 million and an internal rate of return of 19.73% through dilute acid pretreatment, base-catalyzed depolymerization, and PDC fermentation. Capital costs represent a large portion of the total expenses across all scenarios. Revenue from woody feedstocks is largely derived from lignin-based products, while for herbaceous plants, coproducts (fermentable sugars) are the main revenue contributors. The analysis provides insights for the development of lignin valorization biorefineries and guides the chemical industry toward a more sustainable use of renewable carbon sources.

关键词： lignin fermentation biorefinery mixed-integer nonlinear programming superstructure-basedoptimization

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